Maybe, but you could be overcomplicating it.

Does the definition of a(t) suggest that it starts ONLY at t = 0? I'm guessing that it doesn't. In my mind, this function kicks in at the moment cash is deposited and the formula must start counting all over, at that point, for the new deposit.

Remember that simple interest does NOT compound. If you calculate the growth to t = 3 and then calculate the growth to t = 8, you have capitalized the interest earned up to t = 3. That's no good.

8 - 3 =5

3200 = k*(1+0.05*5) -- Find k.